| Exam Board | Edexcel |
|---|---|
| Module | D2 (Decision Mathematics 2) |
| Year | 2002 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Topic | The Simplex Algorithm |
| Type | Interpret optimal tableau |
| Difficulty | Moderate -0.8 This is a straightforward interpretation question requiring students to read an optimal simplex tableau and apply standard decision rules (non-negative profit row entries indicate optimality, read basic variables from the value column, write the profit equation). It involves routine recall and application of simplex algorithm theory with no problem-solving or novel insight required—easier than average A-level maths questions. |
| Spec | 7.07a Simplex tableau: initial setup in standard format7.07b Simplex iterations: pivot choice and row operations7.07c Interpret simplex: values of variables, slack, and objective7.07d Simplex terminology: basic feasible solution, basic/non-basic variable7.07e Graphical interpretation: iterations as edges of convex polygon7.07f Algebraic interpretation: explain simplex calculations |
| Basic variable | \(x\) | \(y\) | \(z\) | \(r\) | \(s\) | \(t\) | Value |
| \(r\) | 0 | 0 | \(1 \frac { 2 } { 3 }\) | 1 | 0 | \(- \frac { 1 } { 6 }\) | \(\frac { 2 } { 3 }\) |
| \(y\) | 0 | 1 | \(3 \frac { 1 } { 3 }\) | 0 | 1 | \(- \frac { 1 } { 3 }\) | \(\frac { 1 } { 3 }\) |
| \(x\) | 1 | 0 | -3 | 0 | -1 | \(\frac { 1 } { 2 }\) | 1 |
| P | 0 | 0 | 1 | 0 | 1 | 1 | 11 |
10. While solving a maximizing linear programming problem, the following tableau was obtained.
\begin{center}
\begin{tabular}{|l|l|l|l|l|l|l|l|}
\hline
Basic variable & $x$ & $y$ & $z$ & $r$ & $s$ & $t$ & Value \\
\hline
$r$ & 0 & 0 & $1 \frac { 2 } { 3 }$ & 1 & 0 & $- \frac { 1 } { 6 }$ & $\frac { 2 } { 3 }$ \\
\hline
$y$ & 0 & 1 & $3 \frac { 1 } { 3 }$ & 0 & 1 & $- \frac { 1 } { 3 }$ & $\frac { 1 } { 3 }$ \\
\hline
$x$ & 1 & 0 & -3 & 0 & -1 & $\frac { 1 } { 2 }$ & 1 \\
\hline
P & 0 & 0 & 1 & 0 & 1 & 1 & 11 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item Explain why this is an optimal tableau.
\item Write down the optimal solution of this problem, stating the value of every variable.
\item Write down the profit equation from the tableau. Use it to explain why changing the value of any of the non-basic variables will decrease the value of $P$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel D2 2002 Q10 [6]}}